Problem 49 - Algebra 2

Question

Psychology Psychologists use an exponential model of the learning process, \(f(t)=c\left(1-e^{-k}\right),\) where \(c\) is the total number of tasks to be learned, \(k\) is the rate of learning, \(t\) is time, and \(f(t)\) is the number of tasks learned. a. Suppose you move to a new school, and you want to learn the names of 30 classmates in your homeroom. If your learning rate for new tasks is 20\(\%\) per day, how many complete names will you know after 2 days? After 8 days? b. Graph the function on your graphing calculator. How many days will it take to learn everyone's name? Explain. c. Open-Ended Does this function seem to describe your own learning rate? If not, how could you adapt it t to reflect your learning rate?

Step-by-Step Solution

Answer

a. After 2 days, you will know approximately 11 names, and after 8 days, approximately 29 names. b. Based on the graph, it will take approximately 10 days to learn everyone's name. c. The answer to this varies by individual, it depends on personal feeling about whether this model represents their learning process, if not, they could tailor the k value to better fit.

Step 1: Calculate Names Learned After 2 and 8 Days

The formula given is \(f(t)=c\left(1-e^{-kt}\right)\), where c is the total tasks to be learned, k is the learning rate, t is the time, and \(f(t)\) is the number of tasks learned. For this scenario, c = 30 (the number of classmates) and k = 0.20 (the learning rate is 20%). Let's find how many names are learned after 2 and 8 days. This means we need to substitute t with 2 and 8 in given formula.

Step 2: Graph the function and estimate the learning time

For this step, we will graph the learning curve \(f(t) = 30 (1 - e^{-0.20t})\) using a graphing calculator. Observing the graph can give an understanding of how quick the learning progresses and can give an estimate for how long it will take to learn all names.

Step 3: Reflect on the model

This step is subjective and it's about reflecting whether this model accurately represents the student's learning rate. If not, one adaptation could be adjusting the k value (learning rate) to better represent the student's personal experience of learning the names.

Key Concepts

Learning Process

The learning process can often be modeled using mathematical functions to predict and analyze how we acquire new information over time. One common model used by psychologists is the exponential model of learning. This model is expressed through the formula \[f(t) = c\left(1-e^{-kt}\right),\]where:

• \(c\) represents the total number of tasks or pieces of information to be learned,
• \(k\) stands for the learning rate, indicating how quickly one learns new tasks,
• \(t\) is the time duration.

\(f(t)\) calculates how many tasks have been learned over time. In real life, the learning process is dynamic and personal; it can be influenced by numerous factors such as exposure, motivation, and individual differences. Understanding the learning process's mathematical underpinnings can provide insights into its efficiency and the time required to achieve mastery in a subject.

Rate of Learning

The rate of learning, denoted by \(k\) in the exponential learning model, essentially describes how efficiently someone can learn new information over time. It is expressed as a percentage, suggesting the proportion of new information or tasks learned per unit of time.
To understand the rate of learning better, imagine you are trying to learn the names of 30 classmates (\(c=30\)). If your learning rate is \(20\%\) per day, the formula shows how many names you would learn over successive days:

• After 2 days: Substitute \(t=2\) into the function to calculate \(f(2) = 30\left(1-e^{-0.40}\right)\).
• After 8 days: Similarly, substitute \(t=8\) to find \(f(8) = 30\left(1-e^{-1.60}\right)\).

Understanding your personal rate of learning can help you adapt strategies that improve how you absorb new information. If you feel your current learning rate does not fit, you can adjust \(k\) to better suit your experience, allowing for more accurate predictions.

Graphing Functions

Graphing functions like the exponential learning model can visually demonstrate how quickly tasks are learned over time. By plotting the function \[f(t) = 30 \left(1 - e^{-0.20t}\right)\] on a graphing calculator, we can observe the learning curve's progression.
A graph typically shows the number of tasks learned (\(f(t)\)) on the y-axis and time (\(t\)) on the x-axis. You'll likely notice that initially, the number of tasks learned increases rapidly but then slows down as you approach learning all 30 tasks.
Graphing helps in several ways:

• Visualizing progress: It becomes easier to see at a glance how your learning unfolds over time.
• Predicting outcomes: The curve will help estimate how quickly you may master all tasks.
• Identifying trends: Changes in learning rate and other variables will alter the curve, which is useful for comparison and analysis.

Making a graph helps you not only understand the theory behind learning models but also provides a tool for practical forecasting of learning outcomes.